{
  "id": "2110.09203",
  "version": "v2",
  "published": "2021-10-18T11:41:18.000Z",
  "updated": "2022-07-10T13:03:58.000Z",
  "title": "Twisted K-theory in motivic homotopy theory",
  "authors": [
    "Elden Elmanto",
    "Denis Nardin",
    "Maria Yakerson"
  ],
  "comment": "corrections in Lemma 5.10",
  "categories": [
    "math.AG",
    "math.AT",
    "math.KT"
  ],
  "abstract": "In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\\mathcal{A}$ over $X$, we construct the $\\mathcal{A}$-twisted motivic spectral sequence, by computing the slices of the motivic twisted algebraic $K$-theory spectrum as a twisted form of motivic cohomology. This generalizes previous results due to Kahn-Levine where $\\mathcal{A}$ is assumed to be pulled back from a base field. Our methods use interaction between the slice filtration and birational geometry. Along the way, we prove a representability result, expressing the motivic space of twisted $K$-theory as an extension of the twisted Grassmannian by the sheaf of \"twisted integers\". This leads to a proof of cdh descent and Milnor excision for twisted homotopy $K$-theory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-10T13:03:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "motivic homotopy theory",
      "twisted k-theory",
      "twisted motivic spectral sequence",
      "study twisted algebraic",
      "smooth variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}